Addon: how to make Blender run check()/redraw when option changed?
Set the filebrowser filename
The code for the ExportHelper mixin class is shown in this question Right way to stop export operator from saving with empty filename?
As demonstrated in question it appears only the FILE_PT_operator is live wired to update the parameters using check.
def update_ext(self, context):
    # if we wish to set filename_ext
    self.filename_ext = f&quot;.{self.extension.lower()}&quot;
    #the filebrowser parameters
    params  = context.space_data.params
    # printed to console
    for k in dir(params):
        print(k, getattr(params, k))
    # set the file name  
    params.filename = f&quot;{Path(self.filepath).stem}.{self.extension.lower()}&quot;   

And use this as an update on the operator enum
extension: bpy.props.EnumProperty(
    name='Extension',
    items=(('FOO', '.foo', '.foo'), ('BAR', '.bar', '.bar')),
    default='FOO',
    update = update_ext,
)

If also using filename_ext make it a property and give it a default to match the enum, however can pretty much do without. The check_existing if set will deal with alerting re pre-existing. (Assume it does haven't checked)
So can pretty much replace the check method to redraw the file browser properties.